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For the past ten years, Michelle has been tracking the average annual rainfall in Boynton Beach, Florida by recording her data in the given table. She has concluded that the relationship can be modeled by a linear function.

Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
Average Rainfall (in inches) 62.33 61.8 61.27 60.74 60.21 59.68 59.15 58.62 58.09 57.56
Use the values provided in the table to create a linear model for the data. (Hint: Let x = 0 represent the year 2004). In your final answer, include all calculations necessary to create the model.
Assuming that the annual amount of rainfall continues to decrease at the same rate, use the linear model created in Part A to predict the rainfall in 2017.

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Answer:

55.44 inches

Step-By-Step Explanation

part a. assuming a perfectly linear relationship, we can find the slope from the first two data points.

slope = m = (change in rainfall)/(change in years)

= (61.80 -62.33)/(2005 -2004) = -0.53/1 = -0.53

then the point-slope form of the equation of the line can be written as

y = m(x -h) +k . . . . . m = -0.53, (h, k) = (0, 62.33)

y = -0.53x +62.33 . . . x = years after 2004

part b. in 2017, x = 2017 -2004 = 13. then the predicted rainfall is

y = -0.53·13 +62.33 = 55.44 . . inches

the predicted rainfall in 2017 is 55.44 in

According to the table given, the linear model for the data is:

[tex]y = -0.53x + 62.33[/tex]

  • Using the linear model, it is found that the predicted rainfall in 2017 is of 55.44 inches.

A linear function is modeled by:

[tex]y = mx + b[/tex]

In which:

  • m is the slope, which is the rate of change, that is, by how much x changes when y changes by 1.
  • b is the y-intercept, which is the value of y when x = 0.

From the table, we have that:

  • We consider x = 0 to be the year 2004, when the average rainfall was of 62.33 inches, hence [tex]b = 62.33[/tex].
  • Each year, the average decreases by 0.53 inches, hence [tex]m = -0.53[/tex].

Hence, the equation is:

[tex]y = -0.53x + 62.33[/tex]

2017 is 13 years after 2004, hence:

[tex]y(13) = -0.53(13) + 62.33 = 55.44[/tex]

The predicted rainfall in 2017 is of 55.44 inches.

You can learn more about linear functions at https://brainly.com/question/16302622

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